#8479031

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,16)(13,22)(14,19)(15,20)(18,23)(21,24)(28,36)(31,37)(35,39)(38,40);(3,7)(5,8)(9,11,15,12)(10,16,20,17)(13,21,23,19)(14,18,24,22)(25,26)(27,34)(28,37)(29,32)(30,33)(31,36)(35,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,17,20,16)(13,19,23,21)(14,18,24,22)(25,27,30,32)(26,29,33,34)(28,35,36,39)(31,38,37,40);(1,3)(2,5)(4,7)(6,8)(9,12,15,11)(10,17,20,16)(13,21,23,19)(14,18,24,22)(25,26)(27,29)(28,31)(30,33)(32,34)(35,38)(36,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,13,15,23)(10,18,20,22)(11,19,12,21)(14,17,24,16)(25,28)(26,31)(27,35)(29,38)(30,36)(32,39)(33,37)(34,40);(9,14,15,24)(10,19,20,21)(11,18,12,22)(13,17,23,16)(25,29)(26,32)(27,33)(28,38)(30,34)(31,39)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386227);